Solution
= Solution
The one-step expected price multiplier under the stated physical <probability> is $1.1p+0.9(1-p)=0.9+0.2p$. Independence lets the expectation of the product factorize, giving
$$
\boxed{\mathbb E_p[S_3]=100(0.9+0.2p)^3.}
$$
The same expression follows by summing the four binomial endpoint <probabilities> times their prices. This is a physical-measure expectation; option replication below uses a separately determined risk-neutral <probability>.